On the probability of being synchronizable

TL;DR

This work establishes that a uniformly random complete deterministic automaton with $n$ states and a fixed non-singleton alphabet is synchronizing with high probability, and it precisely characterizes the convergence rate as $1-Θigl(rac{1}{n}igr)$ for the binary case. The authors develop a probabilistic framework built on stable pairs, cluster structure of random mappings, and highest-branch analysis to prove the lower bound, while a combinatorial upper-bound argument shows that non-synchronization necessitates weak disconnection with probability $Θigl(rac{1}{n}igr)$. They also provide a deterministic algorithm that detects synchronization in linear expected time by exploiting Tarjan's SCC algorithm and the asymptotic structure of random automata, with a safe quadratic fallback for rare cases. The results imply that synchronization is a universal property in random automata rather than an exception and open pathways to understanding minimum-rank behavior and weakly connected regimes in broader alphabets.

Abstract

We prove that a random automaton with $n$ states and any fixed non-singleton alphabet is synchronizing with high probability (modulo an unpublished result about unique highest trees of random graphs). Moreover, we also prove that the convergence rate is exactly $1-Θ(\frac{1}{n})$ as conjectured by [Cameron, 2011] for the most interesting binary alphabet case. Finally, we present a deterministic algorithm which decides whether a given random automaton is synchronizing in linear in $n$ expected time and prove that it is optimal.

Figures (2)

• Figure 1: Left to right: an automaton $\mathfrak{C}_4$, the underlying graphs of its letters $a$ and $b$.
• Figure 2: A digraph with one cycle and a unique highest tree.

Theorems & Definitions (17)

• Conjecture 1: Černý, 1964
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6: Nicaud, 2019
• Theorem 7
• Lemma 9
• Lemma 10
• Lemma 11